College of Engineering
Electrical Engineering Department
Electronics and Communications
Course: EE320 Communications Principles
Text Book: Simon Haykin & Michael Moher,
“Communication Systems”, John Wiley, 5th edition 2010.
Ref Book: Simon Haykin, “Communication
Systems”, John Wiley, 4th edition, 2001.
Associate Prof. Dr. Ahmed Abdelwahab
The signals emitted by information sources and the signals sent over a
transmission channel can be classified into two distinct categories according to
their physical characteristics. These two categories encompass analog and
An analog signal conveys information through a continuous and smooth
variation in time of a physical quantity such as optical, electrical, or acoustical
intensities and frequencies. Well-known analog signals include audio (sound)
and video messages. For example, an electric signal can vary in frequency
(such as the kHz, MHz, GHz designations in radio communications), and its
intensity can range from low to high voltages.
As the signal travels through the channel, various imperfect properties of the
channel induce impairments to the signal. These include electrical noise effects,
signal distortions, and signal attenuation. The function of the receiver is to
extract the weakened and distorted signal from the channel, amplify it, and
restore it as closely as possible to its original signal before transmission and
passing it on to the message destination.
The most fundamental analog signal is the periodic sine wave,
shown in Fig. 1.2. Its three main characteristics are its amplitude,
period or frequency, and phase. The amplitude is the size or
magnitude of the waveform. This is generally designated by the
symbol A and is measured in volts, amperes, or watts, depending
on the signal type. The frequency (designated by f ) is the
number of cycles per second that the wave undergoes (i.e., the
number of times it oscillates per second), which is expressed in
units of hertz (Hz). The period (generally represented by the
symbol T) is the inverse of the frequency, that is, period T = 1/f.
The term phase (designated by the symbol Φ) describes the
position of the waveform relative to time 0, as illustrated in Fig.
1.3. This is measured in degrees or radians (rad):180° = π rad.
A basic sine wave
Two further common characteristics in communications are
the frequency spectrum (or simply spectrum) and the
bandwidth of a signal. The spectrum of a signal is the
range of frequencies that it contains. That is, the spectrum
of a signal is the combination of all the individual sine
waves of different frequencies which make up that signal.
The bandwidth (designated by B) refers to the width of this
Example If the spectrum of a signal ranges from its lowest
frequency flow = 2 kHz to its highest frequency fhigh =22 kHz,
then the bandwidth B = fhigh - flow = 20 kHz.
• Sources of information: Text, speech, audio, pictures,
video and computer data. They are one, two or three
dimensions. Information production and Human
perceptual system for audio and video.
• Communications networks: telephone networks and
• Communications channels: guided propagation such
as: telephone channels (two-wire, coaxial cable or
optical fiber) and free propagation (wireless channels)
such as: broadcasting channels, mobile channels and
In 1837, the telegraph was perfected by Samuel Morse.
In 1875, the telephone was invented by Alexander Graham Bell.
In 1897, A. B. Strowger, devised the automatic step-by-step switch that bears his name;
of all the electromechanical switches devised over the years, the Strowger switch was
the most popular and widely used.
In 1864, James Maxwell formulated the electromagnetic theory of light and predicted
the existence of radio waves.
In 1901, Marconi received a radio signal at Signal Hill in Newfoundland; the radio
signal had originated in Cornwall, England, 1700 miles away across the Atlantic.
In 1904, John Ambrose Fleming invented the vacuum-tube diode, which paved the way
for the invention of the vacuum-tube triode by Lee de Forest in 1906.
In 1918, Edwin H. Armstrong invented the superheterodyne radio receiver; to this day,
almost all radio receivers are of this type.
In 1928, the first all-electronic television system was demonstrated by Philo T.
in 1948, the transistor was invented by Walter H. Brattain, John Bardeen, and William
Shockley at Bell Laboratories. The first silicon integrated circuit (IC) was produced
by Robert Noyce in 1958.
During the period 1943 to 1946, the first electronic digital computer, called the ENIAC,
was built at the Moore School of Electrical Engineering of the University of
Historical Dates (2)
In 1955, John R. Pierce proposed the use of satellites for communications.
In 1966, K. C. Kao and G. A. Hockham of Standard Telephone Laboratories, U.K.,
proposed the use of a clad glass fiber as a dielectric waveguide while the laser had
been invented and developed in 1959 and 1960.
The Advanced Research Project Agency Network (ARPANET), first put into
service in 1971. The development of ARPANET was sponsored by the U.S.
Department of Defense. The pioneering work in packet switching was done on
ARPANET. In 1985, ARPANET was renamed the Internet. The turning point in
the evolution of the Internet occurred in 1990 when Tim Berners-Lee proposed a
hypermedia software interface to the Internet, which he named the World Wide
The spectacular advances in microelectronics, digital computers, and
lightwave systems that we have witnessed to date, and that will
continue into the future, are all responsible for dramatic changes in
the telecommunications environment.
The Modulation Process
The purpose of a communication system is to deliver a
message (information or baseband) signal from an information
source in recognizable form to a user destination, with the
source and the user being physically separated from each
other. To do this, the transmitter modifies the message signal
into a form called passband signal suitable for transmission
over the channel. This modification is achieved by means of a
process known as modulation, which involves varying some
parameter of a carrier wave in accordance with the message
signal. The receiver reconstructs as closely as possible the
original message signal.
This reconstruction is accomplished at the receiver by using a
process known as demodulation, which is the reverse of the
modulation process done at the transmitter.
Types of Modulation
We may classify the modulation process into continuous-wave
modulation and pulse modulation. In continuous-wave (CW) modulation,
a sinusoidal wave is used as the carrier.
When the amplitude of the carrier is varied in accordance with the
message signal, we have amplitude modulation (AM), and when the
angle of the carrier is varied, we have angle modulation. The latter form
of CW modulation may be further subdivided into:
frequency modulation (FM) and phase modulation (PM), in which the
instantaneous frequency and phase of the carrier, respectively, are varied
in accordance with the message signal.
In pulse modulation, on the other hand, the carrier consists of a periodic
sequence of rectangular pulses.
However, owing to the unavoidable presence of noise and distortion in
the received signal, we find that the receiver cannot reconstruct the
original message signal exactly. The resulting degradation in overall
system performance is influenced by the type of modulation scheme
used. Specifically, we find that some modulation schemes are less
sensitive to the effects of noise and distortion than others.
Continuous-Wave (CW) Modulation
Why do we need modulation?
• For use of practical antenna size where the length of the
antenna is proportional to the signal wavelength(λ) about
1/10 of λ. Therefore, the baseband information signal
spectrum needed to be translated up to a higher
frequency band by means of modulations for a smaller
• For use of Multiplexing that is the process of combining
several independent message signals for their
simultaneous transmisson over the same channel.
Multiplexing could be FDM, TDM or CDM or (WDM for
• For use of Better signal-to-noise ratio, it is found that
some modulation schemes are less sensitive to the
effects of noise and distortion than others.
Consider a sinusoidal carrier wave c(t) defined by
where Ac is the carrier amplitude and fc is the carrier frequency
Let m(t) denote the baseband information signal that carries the
specification of the message. The source of carrier wave c(t) is
physically independent of the source responsible for generating
m(t). Amplitude modulation (AM) is defined as a process in
which the amplitude of the carrier wave c(t) is varied about a
mean value, linearly with the baseband signal m(t). An
amplitude-modulated (AM) wave may thus be described, in its
most general form, as a function of time as follows:
where ka is a constant called the amplitude sensitivity of the AM modulator
Note that the envelope of s(t) has essentially the same shape as the baseband
signal m(t) provided that two requirements are satisfied:
1) The amplitude of kam(t) is always less than unity, that is,
Unless the carrier wave becomes overmodulated, resulting in carrier phase
reversals whenever the factor 1 + kam(t) crosses zero. The modulated wave then
exhibits envelope distortion, as in Figure 2.3c. It is therefore apparent that by
avoiding overmodulation, a one-to-one relationship is maintained between the
envelope of the AM wave and the modulating wave for all values of time - a
useful feature, as we shall see later on.
The absolute maximum value of kam(t) multiplied by 100 is referred to as the
2) The carrier frequency fc is much greater than the highest
frequency component B of the message signal m(t), that is fc >> B
Unless an envelope cannot be visualized (and therefore detected) satisfactorily.
The Fourier transform of the AM wave s(t) in
equation 2.2 is given by
Suppose that the baseband signal m(t) is band-
limited to the interval f ≤ B, as in the following
The shape of the spectrum shown in this figure is
intended for the purpose of illustration only.
Spectrum of AM Modulated Wave
Note that the BW of the baseband signal B is referred as W in the above figure.
This spectrum consists of two delta functions weighted by the
factor Ac/ 2 and occurring at ±fc and two versions of the
baseband spectrum translated in frequency by ±fc , and scaled
in amplitude by kaAc /2.
1) As a result of the modulation process, the spectrum of the message
signal m(t) for negative frequencies extending from – B to 0 becomes
completely visible for positive (i.e., measurable) frequencies, provided
that the carrier frequency satisfies the condition fc > B; herein lies the
importance of the idea of "negative" frequencies.
2) For positive frequencies, the portion of the spectrum of an AM wave
lying above the carrier frequency fc is referred to as the upper sideband,
whereas the symmetric portion below fc is referred to as the lower
sideband. For negative frequencies, the upper sideband is represented by
the portion of the spectrum below - fc and the lower sideband by the
portion above –fc. The condition fc > B ensures that the sidebands do not
3) For positive frequencies, the highest frequency component of the AM
wave equals fc + B, and the lowest frequency component equals fc - B.
The difference between these two frequencies defines the transmission
bandwidth BT , for an AM wave, which is exactly twice the message
bandwidth B, that is, BT = 2B.
The greatest advantage of AM: the simplicity of implementation
•In the transmitter, AM is accomplished using a nonlinear device such as
the switching modulator in which the combined sum of the message
signal and carrier wave is applied to a diode, with the carrier amplitude
being large enough to swing across the characteristic curve of the diode.
Fourier analysis of the voltage developed across a resistive load reveals
the generation of an AM component, which may be extracted by means
of a band-pass filter.
•In the receiver, amplitude demodulation is also accomplished using a
nonlinear device Such as a simple and highly effective circuit known as
the envelope detector. The circuit consists of a diode connected in series
with the parallel combination of a capacitor and load resistor. Some
version of this circuit is found in most commercial AM radio receivers.
Provided that the carrier frequency is high enough and the percentage
modulation is less than 100 percent, the demodulator output developed
across the load resistor is nearly the same as the envelope of the
incoming AM wave, hence the name "envelope detector."
AM Major limitations
The transmitted power and the channel bandwidth are the two
primary resources of any communication system, and they should
be used efficiently.
•Amplitude modulation is wasteful of power: The carrier wave c(t) is
completely independent of the information-bearing signal m(t). The
transmission of the carrier wave therefore represents a waste of power,
which means that in amplitude modulation only a fraction of the total
transmitted power is actually affected by m(t).
•Amplitude modulation is wasteful of bandwidth: The upper and lower
sidebands of an AM wave are uniquely related to each other by virtue
of their symmetry about the carrier frequency. This means that as the
transmission of information is concerned, only one sideband is
necessary, and the communication channel therefore needs to provide
only the same bandwidth as the baseband signal. In light of this
observation, amplitude modulation is wasteful of bandwidth as it
requires a transmission bandwidth equal to twice the message
To overcome these limitations, we must make certain
modifications by suppressing the carrier and modifying
the sidebands of the AM wave. These modifications
naturally result in increased system complexity. In
effect, system complexity is traded for improved use of
communication resources. Three types of linear
modulation are introduced to implement such
The demodulated signal vo(t) is therefore proportional to m(t) when the
phase error ϕ is a constant. The amplitude of this demodulated signal is
maximum when ϕ = 0, and it is minimum (zero) when ϕ = ±π/2. The
zero demodulated signal, which occurs for ϕ = ±π/2, represents the
quadrature null effect of the coherent detector. Thus the phase error ϕ in
the local oscillator causes the detector output to be attenuated by a factor
equal to cos ϕ. As long as the phase error ϕ is constant, the detector
provides an undistorted version of the original baseband signal m(t). In
practice, however, we usually find that the phase error ϕ varies randomly
with time, due to random variations in the communication channel. The
result is that at the detector output, the multiplying factor cos ϕ also
varies randomly with time, which is obviously undesirable. Therefore,
provision must be made in the system to maintain the local oscillator in
the receiver in perfect synchronism, in both frequency and phase, with
the carrier wave used to generate the DSB-SC modulated signal in the
transmitter. The resulting system complexity is the price that must be
paid for suppressing the carrier wave to save transmitter power.
A practical synchronous receiver system, suitable for
demodulating DSB-SC waves, is to use the Costas receiver
shown in Figure 2.9.
The quadrature null effect of the coherent detector may also be put to good use
in the construction of the so-called quadrature-carrier multiplexing or
Quadrature Amplitude Modulation
• This scheme enables two DSB-SC modulated waves (resulting from
the application of two physically independent message signals) to
occupy the same channel bandwidth, and then it allows for the
separation of the two message signals at the receiver output. It is
therefore a bandwidth-conservation scheme.
• To maintain synchronization between transmitter and receiver, a
pilot signal outside the passband of the modulated signal may be sent
. In this method, the pilot signal typically consists of a low-power
sinusoidal tone whose frequency and phase are related to the carrier
wave c(t); at the receiver, the pilot signal is extracted by means of a
suitably tuned circuit and then translated to the correct frequency for
use in the coherent detector.
A modulated wave s1 (t)
whose spectrum is centered
on a carrier frequency f1 ,
and the requirement is to
translate it upward in
frequency such that its
carrier frequency is changed
from f1 to a new value f2.
This requirement may be
accomplished using the
mixer shown in Figure 2.16.
Specifically, the mixer is a
device that consists of a
product modulator followed
by a band-pass filter.
The spectrum S/( f ) of the resulting signal s/(t) at the product
modulator output is shown in Figure 2.17b.
The signal s/(t) may be viewed as the sum of two modulated
components: one component represented by the shaded spectrum and
the other component represented by the no shaded spectrum in the
figure. Depending on whether the incoming carrier frequency f1 is
translated upward or downward. Two different situations may be
identified, as following:
Up conversion, where the translated carrier frequency f2 is greater
than the incoming carrier frequency f1, and the required local oscillator
frequency fLO is therefore defined by
f2 = fLO +f1
i.e. fLO = f2 – f1
Down conversion, where the translated carrier frequency f2
is smaller than the incoming carrier frequency f1, and the
required oscillator frequency fLO is therefore defined by
fLO = f1 – f2
The mixer is now referred to as a frequency-down converter.
Note that in this case the translated carrier frequency f2 has
to be larger than B (i.e., one half of the bandwidth of the
modulated signal) to avoid sideband overlap. It is important
to note that mixing is a linear operation. Accordingly, the
relation of the sidebands of the incoming modulated wave to
the carrier is completely preserved at the mixer output.
the superheterodyne receiver consists of a radio-frequency
(RF) section, a mixer and local oscillator, an intermediate-
frequency (IF) section, demodulator, and power amplifier.
Figure 2.32 shows the block diagram of a superheterodyne
receiver for amplitude modulation using an envelope
detector for demodulation.
The incoming amplitude-modulated wave is picked up by the
receiving antenna and amplified in the RF section which is
tuned to the carrier frequency of the incoming wave.
The combination of mixer and local oscillator (of adjustable
frequency) provides a heterodyning function, whereby the
incoming signal is converted to a predetermined fixed
intermediate frequency, usually lower than the incoming
carrier frequency. This frequency translation is achieved
without disturbing the relation of the sidebands to the carrier;
The result of the heterodyning is to produce an intermediate-
frequency carrier defined by
In a superheterodyne receiver the mixer will develop an intermediate frequency output
when the input signal frequency is greater or less than the local oscillator frequency by
an amount equal to the intermediate frequency. i.e., there are two input frequencies,
namely, |fLO ± fIF| which will result in fIF at the mixer output. This introduces the
possibility of simultaneous reception of two signals differing in frequency by twice the
For example, a receiver tuned to 0.65 MHz and having an IF of 0.455 MHz is
subject to an image interference at 1.56 MHz; indeed, any receiver with this
value of IF, when tuned to any station, is subject to image interference at a
frequency of 0.910 MHz higher than the desired station.
Since the function of the mixer is to produce the difference between two
applied frequencies, it is incapable of distinguishing between the desired
signal and its image frequency. The only practical cure for image
interference is to employ highly selective stages in the RF section (i.e.,
between the antenna and the mixer) in order to favor the desired signal
and discriminate against the undesired or image signal.
The effectiveness of suppressing unwanted image signals increases as the
number of selective stages in the RF section increases, and as the ratio of
intermediate to signal frequency increases.
The IF section consists of one or more amplification stages at
fixed tuned IF frequency, with a bandwidth corresponding to
that required for the particular type of modulation that the
receiver is intended to handle (10 KHz for AM). The IF
section provides most of the amplification and selectivity in
IF section can effectively suppress adjacent channel
interference because of its high selectivity which can not be
achieved in RF section where it is difficult to design a tunable
sharp BPF of BW = 10KHz with center frequency at radio
The output of the IF section is applied to a demodulator, the
purpose of which is to recover the baseband signal. If coherent
detection is used, then a coherent signal source must be
provided in the receiver. The final operation in the receiver is
the power amplification of the recovered message signal.
In single-sideband modulation, only the upper or lower
sideband is transmitted. SSB modulated wave may be
generated by using the frequency discrimination
method that consists of a product modulator, which
generates a DSB-SC modulated wave, and a band-pass
filter, which is designed to pass one of the sidebands of
this modulated wave and suppress the other. For the
generation of an SSB modulated signal to be possible,
the message spectrum must have an energy gap
centered at the origin.
This requirement is naturally satisfied by voice signals, whose energy gap is
about 600 Hz wide (i.e.it extends from -300 to +300 Hz). Thus, assuming that the
upper sideband is retained, the spectrum of the SSB modulated signal is as shown in
Figure 2.11b. In designing the band-pass filter used in the frequency-discriminator
for generating a SSB-modulated wave, the use of highly selective filters, which can
only be realized in practice by means of crystal resonators are required. Moreover,
the following three basic requirements must be met:
• The desired sideband lies inside the passband of the filter.
• The unwanted sideband lies inside the stopband of the filter.
• The filter's transition band, which separates the passband from the stopband, is
twice the lowest frequency component of the message signal.
To demodulate a SSB modulated signal s(t), we may use a coherent detector,
which multiplies s(t) by a locally generated carrier and then low-pass filters the
product. This method of demodulation assumes perfect synchronism between the
oscillator in the coherent detector and the oscillator used to supply the carrier
wave in the transmitter.
This requirement is usually met in one of two ways:
•A low-power pilot carrier is transmitted in addition to the selected sideband.
•A highly stable oscillator, tuned to the same frequency as the carrier frequency,
is used in the receiver.
In the latter method, it is unavoidable that there would be some phase error ϕ in
the local oscillator output with respect to the carrier wave used to generate the
incoming SSB modulated wave. The effect of this phase error is to introduce a
phase distortion in the demodulated signal, where each frequency component of
the original message signal undergoes a constant phase shift ϕ. This phase
distortion is tolerable in voice communications because the human ear is
relatively insensitive to phase distortion. In particular, the presence of phase
distortion gives rise to a Donald Duck voice effect. In the transmission of music
and video signals, on the other hand, the presence of this form of waveform
distortion is absolutely unacceptable.
Multiplexing is another important signal processing operation,
whereby a number of independent signals can be combined into
a composite signal suitable for transmission over a common
channel. Voice frequencies transmitted over telephone systems,
range from 300 to 3100 Hz.
To transmit a number of these signals over the same channel,
the signals must be kept apart so that they do not interfere with
each other, and thus they can be separated at the receiving end.
This is accomplished by separating the signals either in
frequency or in time.
The technique of separating the signals in frequency is referred
to as frequency-division multiplexing (FDM), whereas the
technique of separating the signals in time is called time-
division multiplexing (TDM).
The input voice signal is first applied to a low-pass filter, which is designed to remove
high-frequency components that do not contribute significantly to signal representation
but are capable of disturbing other message signals that share the common channel.
The filtered signals are then applied to modulators (usually SSB modulators) that shift the
frequency ranges of the signals so as to occupy mutually exclusive frequency intervals. The
necessary carrier frequencies needed to perform these frequency translations are obtained from a
In practice, each voice input is usually assigned a bandwidth
of 4 kHz. The band-pass filters following the modulators are
used to restrict the band of each modulated wave to its
prescribed range. The resulting bandpass filter outputs are next
combined in parallel to form the input to the common channel.
At the receiving terminal, a bank of band-pass filters, with
their inputs connected in parallel, is used to separate the
message signals on a frequency-occupancy basis. Finally, the
original message signals are recovered by individual
Note that the FDM system shown in Figure 2.18 operates in
only one direction. To provide for two-way transmission, as in
telephony, for example, we have to completely duplicate the
multiplexing facilities, with the components connected in
reverse order and with the signal waves proceeding from right
practical implementation of FDM system
The practical implementation of an FDM system usually involves many
steps of modulation and demodulation, as illustrated in Figure 2.19. The
first multiplexing step combines 12 voice inputs into a basic group,
which is formed by having the nth input modulate a carrier at frequency
fn = 60 + 4n kHz, where n = 1,2, . . . , 12. The lower sidebands are then
selected by band-pass filtering and combined to form a group of 12 lower
sidebands (one for each voice input). Thus the basic group occupies the
frequency band 60 to 108 kHz. The next step in the FDM hierarchy
involves the combination of five basic groups into a supergroup. This is
accomplished by using the nth group to modulate a carrier of frequency
fn = 372 + 48n kHz, where n = 1,2, . . . , 5 . Here again the lower
sidebands are selected by filtering and then combined to form a
supergroup occupying the band 312 to 552 kHz. Thus a supergroup is
designed to accommodate 60 independent voice inputs. In a similar
manner, supergroups are combined into mastergroups, and mastergroups
are combined into very large groups.
2.6 -Angle Modulation
Angle modulation in which the angle of the carrier wave is
varied according to the baseband information signal. In this
method of modulation, the amplitude of the carrier wave is
maintained constant. An important feature of angle modulation
is that it can provide better discrimination against noise and
interference than amplitude modulation. However, this
improvement in performance is achieved at the expense of
increased transmission bandwidth; that is, angle modulation
provides us with a practical means of exchanging channel
bandwidth for improved noise performance. Such a tradeoff
is not possible with amplitude modulation, regardless of its
Let θi (t) denote the instantaneous angle of a
modulated sinusoidal carrier, assumed to be a function
of the message signal. The resulting angle-modulated
wave can be expressed as
If θi (t) increases monotonically with time, the average
frequency in Hertz, over an interval from t to t + Δt, is
Where fi(t) defines the instantaneous frequency of the angle-modulated signal s(t). The
angle-modulated signal s(t) may be interpreted as a rotating phasor of length Ac and
angle θi (t).
The angular velocity of such a phasor is dθi(t)/dt measured in radians per second.
There are two commonly used methods, phase modulation and frequency modulation,
in which the angle θ,(t) may be varied in some manner with the message (baseband)
signal, defined as follows:
Phase modulation (PM) is that form of angle modulation in which the angle θi (t) is
varied linearly with the message signal m(t), as shown by
The term 2πfct represents the angle of the unmodulated carrier; and the constant kp
represents the phase sensitivity of the modulator, expressed in radians per volt on the
assumption that m(t) is a voltage waveform. The phase-modulated signal s(t) is thus
described in the time domain by
Frequency modulation (FM) is that form of angle
modulation in which the instantaneous frequency fi (t)
is varied linearly with the message signal m(t), as
The term fc represents the frequency of the
unmodulated carrier, and the constant kf represents the
frequency sensitivity of the modulator, expressed in
Hertz per volt. Therefore, the instantaneous phase can
be expressed as
The frequency-modulated signal is therefore described
in the time domain by
The envelope of a PM or FM signal is constant (equal
to the carrier amplitude), whereas the envelope of an
AM signal is dependent on the message signal.
All the properties of PM signals can be deduced from those of FM
signals and vice versa. Henceforth, we concentrate our attention on
The FM signal s(t) is a nonlinear function of the modulating signal m(t),
which makes frequency modulation a nonlinear modulation process.
The simplest case possible, namely, that of a single-tone modulation that
produces a narrowband FM signal is first considered. Hence, Consider a
sinusoidal modulating signal defined by
The instantaneous frequency of the resulting FM signal equals
The quantity Δf is called the frequency deviation, representing the
maximum departure of the instantaneous frequency of the FM signal
from the carrier frequency fc. A fundamental characteristic of an FM
signal is that the frequency deviation Δf is proportional to the amplitude
of the modulating signal and is independent of the modulation frequency.
In a physical sense, the parameter β represents the phase
deviation of the FM signal, that is, the maximum departure of
the angle θi(t). from the angle 2πfct of the unmodulated carrier;
hence, β is measured in radians. The FM signal itself is given
Depending on the value of the modulation index β, we may
distinguish two cases of frequency modulation:
•Narrowband FM, for which β is small compared to one radian.
•Wideband FM, for which β is large compared to one radian.
Assuming that the modulation index β is small compared to one radian, we may use
the following approximations:
Equation (2.35) may be expanded as follows:
This expression is somewhat similar to the corresponding one defining
an AM signal, which is as follows:
In the case of sinusoidal modulation, the
basic difference between an AM signal
and a narrowband FM signal is that the
algebraic sign of the lower side frequency
in the narrowband FM is reversed. Thus,
a narrowband FM signal requires
essentially the same transmission
bandwidth (i.e.,2fm) as the AM signal.
Using the complex representation of band-pass signals
described in Appendix 2 of the textbook. Specifically, assume
that the carrier frequency fc is large enough (compared to the
bandwidth of the FM signal) to justify rewriting equation
(2.33) in the form
This is the desired form for the Fourier series representation of
the single-tone FM signal s(t) for an arbitrary value of β.
is called the nth order Bessel function of the first kind and
argument β which are plotted in figure 2.23 for orders
0,1,2,3 & 4 versus the modulation index β.
For small values of the modulation index β, we have
• The spectrum of an FM signal contains a carrier component and an infinite set of side
frequencies located symmetrically on either side of the carrier at frequency
separations of fm , 2fm , 3fm, - . . . In this respect, the result is unlike that which
prevails in an AM system, since in an AM system a sinusoidal modulating signal
gives rise to only one pair of side frequencies.
• For the special case of β small compared with unity, only the Bessel coefficients Jo(β)
and J1(β) have significant values, so that the FM signal is effectively composed of a
carrier and a single pair of side frequencies at fc ± fm , . This situation corresponds to
the special case of narrowband FM that was considered earlier.
• The amplitude of the carrier component varies with β according to Jo(β). That is,
unlike an AM signal, the amplitude of the carrier component of an FM signal is
dependent on the modulation index β. The physical explanation for this property is
that the envelope of an FM signal is constant, so that the average power of such a
signal developed across a 1-ohm resistor is also constant, as shown by
When the carrier is modulated to generate the FM signal, the power in the side
frequencies may appear only at the expense of the power originally in the carrier.
The average power of an FM signal may also be determined from Equation (2.48),
An approximate rule for the transmission bandwidth of
an FM signal generated by a single-tone modulating
signal of frequency fm as follows:
This empirical relation is known as Carson's rule.
A definition based on retaining the maximum
number of significant side frequencies whose
amplitudes are all greater than some selected
Value may be used for the FM transmission
bandwidth. A convenient choice for this value
is 1 % of the unmodulated carrier amplitude.
Consider next the more general case of an arbitrary modulating
signal m(t) with its highest frequency component denoted by B.
The bandwidth required to transmit an FM signal generated by this
modulating signal is estimated by using a worst-case tone
modulation analysis. Specifically, we first determine the so-called
deviation ratio D, defined as the ratio of the frequency deviation
Δf, which corresponds to the maximum possible amplitude of the
modulation signal m(t), to the highest modulation frequency B;
these conditions represent the extreme cases possible. The
deviation ratio D plays the same role for nonsinusoidal modulation
that the modulation index β plays for the case of sinusoidal
modulation. Then, replacing β by D and replacing fm with B, we
may use Carson's rule given by Equation (2.55) or the universal
curve of Figure 2.26 to obtain a value for the transmission
bandwidth of the FM signal. From a practical viewpoint, Carson's
rule somewhat underestimates the bandwidth requirement of an FM
system, whereas using the universal curve of Figure 2.26 yields a
somewhat conservative result. Thus, the choice of a transmission
bandwidth that lies between the bounds provided by these two rules
of thumb is acceptable for most practical purposes.
In North America, the maximum value of frequency deviation
Δf is fixed at 75 kHz for commercial FM broadcasting by
radio. If we take the modulation frequency B = 15 kHz, which
is typically the "maximum" audio frequency of interest in FM
transmission, we find that the corresponding value of the
deviation ratio is
Using Carson's rule, replacing β by D, and replacing fm by B,
the approximate value of the transmission bandwidth of the
FM signal is obtained as
On the other hand, use of the curve of Figure 2.26 gives the
transmission bandwidth of the FM signal to be
In practice, a bandwidth of 200 kHz is allocated to each FM
There are essentially two basic methods of generating
frequency-modulated signals, namely, direct FM and
In the direct method the carrier frequency is directly
varied in accordance with the input baseband signal,
which is readily accomplished using a voltage-controlled
In the indirect method, the modulating signal is first used
to produce a narrowband FM signal, and frequency
multiplication is next used to increase the frequency
deviation to the desired level. The indirect method is the
preferred choice for frequency modulation when the
stability of carrier frequency is of major concern as in
commercial radio broadcasting.
The indirect method of generating a
wideband FM signal
The use of crystal controlled oscillator provides frequency stability. To minimize the
distortion inherent in the phase modulator, the maximum phase deviation (modulation
index β) is kept small, thereby resulting in a narrowband FM signal. The implementation
of the narrow-band phase modulator is described in Figure 2.21. The narrowband FM
signal is next multiplied in frequency by means of a frequency multiplier so as to produce
the desired wideband FM signal.
It consists of a nonlinear device followed by a band-pass filter, as shown above.
The implication of the nonlinear device being memoryless is that it has no
The memoryless nonlinear device is an nth power-law device. The mid-band
frequency of the band-pass filter in Figure 2.28 is set equal to nfc where fc is the
carrier frequency of the incoming FM signal s(t). Moreover, the bandpass filter
is designed to have a bandwidth equal to n times the transmission bandwidth of s(t).
The input instantaneous frequency of the input FM wave
The output instantaneous frequency of the output FM wave will be
Frequency demodulation is the process that enables us to
recover the original modulating signal from a frequency-
modulated signal. The objective is to produce a transfer
characteristic that is the inverse of that of the frequency
modulator, which can be realized directly or indirectly. The
direct method of frequency demodulation involves the
use of a popular device known as a frequency discriminator,
whose instantaneous output amplitude is directly proportional
to the instantaneous frequency of the input FM signal.
The indirect method of frequency demodulation uses
another popular device known as a phase-locked loop.
Balanced frequency discriminator
The ideal frequency discriminator may be modeled as a
pair of slope circuits followed by envelope detectors and
finally a summer, as in Figure 4.14a. This scheme is
called a balanced frequency discriminator which can be
closely realized using the circuit shown in Figure 4.14b.
The upper and lower resonant filter sections of this circuit
are tuned to frequencies above and below the
unmodulated carrier frequency fc, respectively.
In Figure 4.14c the amplitude responses of these two
tuned filters are plotted , together with their total
response, assuming that both filters have a high Q-factor.
The Q-factor is equal to the resonant frequency
divided by the 3-dB bandwidth of the circuit. In
the RLC parallel resonant circuits shown in
Figure 4.14b, the resistance R is contributed
largely by imperfections in the inductive
elements of the circuits.
Stereo multiplexing is a form of frequency-division
multiplexing (FDM) designed to transmit two separate signals
via the same carrier. It is widely used in FM radio
broadcasting to send two different elements of a program
(e.g., two different sections of an orchestra, a vocalist and an
accompanist) so as to give a spatial dimension to its perception
by a listener at the receiving end.
The specification of standards for FM stereo transmission is
influenced by two factors:
1. The transmission has to operate within the allocated FM
2. It has to be compatible with monophonic radio receivers.
The Block Diagram of the FM Stereo Multiplexing
Let ml(t) and mr(t) denote the signals picked
up by left-hand and right-hand microphones
at the transmitter. The sum signal is left
unprocessed in its baseband form; it is
available for monophonic reception. The
difference signal and a 38-kHz subcarrier
(derived from a 19-kHz crystal oscillator by
frequency doubling) are applied to a product
modulator, thereby producing a DSB-SC
modulated wave. In addition to the sum
signal and this DSB-SC modulated wave, the
multiplexed signal m(t) also includes a 19-
kHz pilot to provide a reference for the
coherent detection of the difference signal at
the stereo receiver. Thus the multiplexed
signal is described as shown below where fc
= 19 kHz, and K is the amplitude of the pilot
FM Radio Receiver
The block diagram of a superheterodyne receiver for
frequency modulation is exactly as that of the AM
explained in Figure 2.32 except for the FM receiver, a
frequency discriminator is used for demodulation.
Typical frequency parameters of commercial AM and
FM radio receivers are listed in Table 2.3 above.
In continuous-wave (CW) modulation, some parameter of a sinusoidal carrier wave is
varied continuously in accordance with the message signal.
In pulse modulation, some parameter of a pulse train is varied in accordance with the
There are two families of pulse modulation: analog pulse modulation and digital pulse
modulation. In analog pulse modulation, a periodic pulse train is used as the carrier
wave, and some characteristic feature of each pulse (e.g., amplitude, duration, or
position) is varied in a continuous manner in accordance with the corresponding
sample value of the message signal. Thus in analog pulse modulation, information is
transmitted basically in analog form, but the transmission takes place at discrete times.
In digital pulse modulation, on the other hand, the message signal is represented in a
form that is discrete in both time and amplitude, thereby permitting its transmission
in digital form as a sequence of coded pulses; this form of signal transmission has no
The use of coded pulses for the transmission of analog information-bearing signals
represents a basic ingredient in the application of digital communications. This chapter
may therefore be viewed as a transition from analog to digital communications in
study of the principles of communication systems. We begin the discussion by
describing the sampling process, which is basic to all pulse modulation systems,
whether they are analog or digital.
The sampling process
The sampling process is usually described in
the time domain. Through use of the sampling
process, an analog signal is converted into a
corresponding sequence of samples that are
usually spaced uniformly in time. Clearly, for
such a procedure to have practical utility, it is
necessary that we choose the sampling rate
properly, so that the sequence of samples
uniquely defines the original analog signal. This is
the essence of the sampling theorem, which is
derived in what follows.
Where Ts is the sampling period, and its reciprocal fs = 1ITs is the sampling rate. This
ideal form of sampling is called instantaneous sampling.
where G ( f ) is the Fourier transform of the original signal g(t) of finite energy, which is
specified for all time. Equation (3.2) states that the process of uniformly sampling a
continuous-time signal of finite energy results in a periodic spectrum with a period equal to
the sampling rate.
The Sampling Theorem
The sampling theorem for strictly band-limited signals of finite energy
may be stated in two equivalent parts, which apply to the transmitter
and the receiver of a pulse modulation system, respectively:
• A band-limited signal of finite energy, which has no frequency
components higher than B Hertz, is completely described by
specifying the values of the signal at instants of time separated by
• A band-limited signal of finite energy, which has no frequency
components higher than B Hertz, may be completely recovered from a
knowledge of its samples taken at the rate of 2B samples per second.
• The sampling rate of 2 B samples per second, for a signal whose
bandwidth of B Hertz, is called the Nyquist rate; its reciprocal 1/2 B
(measured in seconds) is called the Nyquist interval.
some aliasing is produced by the
sampling process if the sampling
frequency is less than Nyquist rate.
Aliasing refers to the phenomenon
of a high-frequency component in
the spectrum of the signal
seemingly taking on the identity of a
lower frequency in the spectrum of
its sampled version, as illustrated in
To combat the effects of aliasing in practice, we may use two corrective measures, as
1. Prior to sampling, a low-pass anti-aliasing filter is used to attenuate those high
frequency components of the signal that are not essential to the information being
conveyed by the signal.
2. The filtered signal is sampled at a rate slightly higher than the Nyquist rate.
The reconstruction filter is a low-pass filter with
• A passband extending from - W to W, which is itself
determined by the anti-aliasing filter.
• A transition band extending (for positive frequencies)
from W to fs - W, where fs is the sampling rate.
The fact that the reconstruction filter has a
well-defined transition band means that it is
In pulse-amplitude modulation (PAM), the amplitudes of
regularly spaced pulses are varied in proportion to the
corresponding sample values of a continuous
message signal; the pulses can be of a rectangular form or
some other appropriate shape.
The dashed curve in fig.3.5
depicts the waveform of
a message signal m(t),
and the sequence of amplitude-modulated rectangular pulses
shown as solid lines represents the corresponding PAM signal
In digital circuit technology, two operations that are jointly
called "sample and hold" involve in the generation of the
PAM signal. One important reason for intentionally
lengthening the duration of each sample is to avoid the use of
an excessive channel bandwidth, since bandwidth is inversely
proportional to pulse duration T. However, care has to be
exercised in how long we make the sample duration T. In
order to recover (reconstruct) the original message signal m(t),
The PAM signal s(t) is passed through a LPF whose frequency
response is defined in Figure 3.4c followed by an equalizer in
order to compensate for the amplitude distortion. However,
for a duty cycle T/Ts ≤ 0.1, the amplitude distortion is less than
0.5 percent, in which case the need for equalization may be
Time Division Multiplexing (TDM)
An important feature of the sampling process is a
conservation of time. That is, the transmission of the
message samples engages the communication channel
for only a fraction of the sampling interval on a
periodic basis, and in this way some of the time
interval between adjacent samples is cleared for use by
other independent message sources on a time-shared
basis resulting in a time-division multiplex (TDM)
system, which enables the joint utilization of a
common communication channel by a plurality of
independent message sources without mutual
interference among them.
Synchronization is essential for a satisfactory operation of the TDM system. The way
this synchronization is implemented depends naturally on the method of pulse
modulation used to transmit the multiplexed sequence of samples.
The TDM system is highly sensitive to dispersion in the common channel, that is, to
variations of amplitude with frequency or lack of proportionality of phase with
Accordingly, accurate equalization of both magnitude and phase responses of the
channel is necessary to ensure a satisfactory operation of the TDM system
TDM is immune to nonlinearities in the channel as a source of cross-talk. Because
different message signals are not simultaneously applied to the channel.
The Quantization Process
The sampling process takes care of the discrete-time
representation of the message signal while the quantization
process takes care of the discrete amplitude representation of
the message signal. A continuous signal, such as voice, has a
continuous range of amplitudes and therefore its samples have
a continuous amplitude range. In other words, within the finite
amplitude range of the signal, we find an infinite number of
amplitude levels. It is not necessary in fact to transmit the
exact amplitudes of the samples. Any human sense (the ear or
the eye), as ultimate receiver, can detect only finite intensity
differences. This means that the original continuous signal
may be approximated by a signal constructed of discrete
amplitudes selected on a minimum error basis from an
The amplitude quantization is defined as the process
of transforming the sample amplitude m(nTs) of a
message signal m(t) at time t = nTs into a discrete
amplitude v(nTs) taken from a finite set of possible
amplitudes. the quantization process is assumed
memoryless and instantaneous, which means that the
transformation at time t = nTs is not affected by earlier
or later samples of the message signal. This simple
form of scalar quantization, though not optimum, is
commonly used in practice. The signal amplitude m is
specified by the index k if it lies inside the partition
where L is the total number of amplitude levels used
in the quantizer. Note that the notation is simplified by
dropping the time index.
The discrete amplitudes mk,
k = 1, 2,. . . , L, at the quantizer
input are called decision levels
or decision thresholds. At the
quantizer output, the index k is
transformed into an amplitude vk
that represents all amplitudes of
the cell ; the discrete
amplitudes vk, k = 1,2,. . . , L,are
called representation levels or
reconstruction levels, and the
spacing between two adjacent
representation levels is called a
quantum or step-size. Thus, the
quantizer output v equals
vk if the input signal sample m
belongs to the interval
Quantizers can be of a uniform or nonuniform type. In a uniform quantizer, the
representation levels are uniformly spaced; otherwise, the quantizer is nonuniform.
The use of quantization introduces
an error defined as the difference
between the input signal m and the
output signal v. The error is called
quantization noise. Figure 3.11
illustrates a typical variation of the
quantization noise as a function of
time, assuming the use of a uniform
quantizer of the midtread type.
The quantizer input m can always
be assumed to be a sample value
of a zero-mean random variable M.
(If the input has a nonzero mean, it
can always be removed by
subtracting the mean from
the input and then adding it back
With the input M having zero mean, and the quantizer
assumed to be symmetric as in Figure 3.10, it follows
that the quantizer output V and therefore the
quantization error Q, will also have zero mean.
Consider an input m of continuous amplitude in the
range (-mmax, mmax). Assuming a uniform quantizer of
the midrise type illustrated in Figure 3.10b, the step-
size of the quantizer is given by &
where L is the total number of representation levels
and R denote the number of bits per sample used in the
construction of the binary code. The probability
density function of the quantization error Q is assumed
to be uniform as follows:
Pulse Code Modulation (PCM)
In pulse-code modulation (PCM), a message signal is
represented in discrete form in both time and
amplitude. This form of signal representation permits
the transmission of the message signal as a sequence
of coded binary pulses. Given such a sequence, the
effect of channel noise at the receiver output can be
reduced to a negligible level simply by making the
average power of the transmitted binary PCM wave
large enough compared to the average power of the
Standard Telephone Speech Quantization
The range of voltages covered by voice signals, from the peaks of loud talk to the
weak passages of weak talk, is on the order of 1000 to 1. By using a nonunifom
quantizer with the feature that the step-size increases as the separation from the
origin of the input-output amplitude characteristic is increased, the large end steps
of the quantizer can take care of possible departures of the voice signal into the
large amplitude ranges that occur relatively infrequently. In other words, the weak
passages, which need more protection, are favored at the expense of the loud
In this way, a nearly uniform percentage precision is achieved throughout the
greater part of the amplitude range of the input signal, with the result that fewer
steps are needed than would be the case if a uniform quantizer were used.
The use of a nonuniform quantizer is equivalent to passing the baseband signal
through a compressor and then applying the compressed signal to a uniform
quantizer. There are tow particular forms of compression law that is used in
practice are: µ law and A law which are defined respectively by
To restore the signal samples
to their correct relative level,
a device in the receiver with a
to the compressor is used.
Such a device is called an
expander. The compression
and expansion laws are
exactly inverse so that, except
for the effect of quantization,
the expander output is equal
to the compressor input. The
combination of a compressor
and an expander is called a
Where m and v are the normalized input and output voltages, µ and A are positive
constants. The typical values used in practice are: µ = 255 and A = 87.6. It is also of
interest to note that in actual PCM systems, the companding circuitry does not produce an exact
replica of the nonlinear compression curves shown in Figure 3.14. Rather, it provides a piecewise
linear approximation to the desired curve. By using a large enough number of linear segments,
the approximation can approach the true compression curve very closely.
To exploit the advantages of sampling and quantizing for the
purpose of making the transmitted signal more robust to noise,
interference and other channel impairments, we require the use
of an encoding process to translate the discrete set of sample
values to a more appropriate form of signal. A particular
arrangement of symbols used in a code to represent a single
value of the discrete set is called a codeword or character. The
two symbols of a binary code are customarily denoted as 0 and
1. in a binary code, each codeword consists of R bits. Thus R
denotes the number of bits per sample. Then, using such a
code, we may represent a total of 2R distinct samples. For
example, a sample quantized into one of 256 levels may be
represented by an 8-bit codeword.
(a) In unipolar NRZ signaling, symbol 1 is
represented by transmitting a pulse of
amplitude A for the
duration of the symbol, and symbol 0 is
represented by switching off the pulse (on-off
signaling). Disadvantages of on-off signaling
are the waste of power due to the transmitted
DC level and the fact that the power spectrum
of the transmitted signal does not approach zero
at zero frequency.
(b) in Polar NRZ signaling, symbols 1 and 0
are represented by transmitting pulses of
amplitudes +A and -A, respectively. This line
code is relatively easy to generate but its
disadvantage is that the power spectrum of the
signal is large near zero frequency.
(c) in unipolar RZ signaling, symbol 1 is represented by a rectangular
pulse of amplitude A and half-symbol width, and symbol 0 is represented
by transmitting no pulse. An attractive feature of this line code is the
presence of delta functions at f = 0, ±l/Tb in the power spectrum of the
transmitted signal, which can be used for bit timing recovery at the
receiver. However, its disadvantage is that it requires 3 dB more power
than polar return-to-zero signaling for the same probability of symbol
(d) in bipolar RZ signaling, three amplitude levels are used. Specifically,
positive and negative pulses of equal amplitude (i.e., +A and -A) are
used alternately for symbol 1, with each pulse having a half-symbol
width; no pulse is always used for symbol 0. A useful property of the
BRZ signaling is that the power spectrum of the transmitted signal has no
DC component and relatively insignificant low-frequency components
for the case when symbols 1 and 0 occur with equal probability.
(e) Split-phase (Manchester code)
In this method of signaling, symbol 1 is represented by a positive pulse
of amplitude A followed by a negative pulse of amplitude -A, with both
pulses being half-symbol wide. For symbol 0, the polarities of these two
pulses are reversed. The Manchester code suppresses the DC component
and has relatively insignificant low-frequency components, regardless of
the signal statistics. This property is essential in some applications.
The most important feature of PCM systems lies in the ability to control the effects of distortion
and noise produced by transmitting a PCM signal through a channel. This capability is
accomplished by reconstructing the PCM signal by means of a chain of regenerative repeaters
located at sufficiently close spacing along the transmission route.
The equalizer shapes the received pulses so as to compensate for the effects of amplitude and
phase distortions produced by the no ideal channel. The timing circuitry provides a periodic
pulse train, derived from the received pulses, for sampling the equalized pulses at the instants at
time where the signal-to-noise ratio is a maximum. Each sample so extracted is compared to a
predetermined threshold in the decision-making device. In each bit interval, a decision is then
made whether the received symbol is a 1 or a 0 on the basis of whether the threshold is exceeded
or not. If the threshold is exceeded, a clean new pulse representing symbol 1 is transmitted to the
next repeater. Otherwise, another clean new pulse representing symbol 0 is transmitted.
In practice, however, the regenerated signal departs from
the original signal for two main reasons:
1. The unavoidable presence of channel noise and
interference causes the repeater to make wrong decisions
occasionally, thereby introducing bit errors into the
2. If the spacing between received pulses deviates from its
assigned value, a jitter is introduced into the regenerated
pulse position, thereby causing distortion.
• Decoding: The first operation in the receiver is to
regenerate (i.e., reshape and clean up) the received pulses
one last time. These clean pulses are then regrouped into
code words and decoded (i.e., mapped back) into a
quantized PAM signal.
• Filtering: The final operation in the receiver is to recover
the message signal by passing the decoder output through a
low-pass reconstruction filter whose cutoff frequency is
equal to the message bandwidth B. Assuming that the
transmission path is error free, the recovered signal
includes no noise with the exception of the initial
distortion introduced by the quantization process.
Noise Considerations in PCM Systems
The performance of a PCM system is influenced by
two major sources of noise:
1. Channel noise, which is introduced anywhere
between the transmitter output and the receiver
input. Channel noise is always present, once the
equipment is switched on.
2. Quantization noise, which is introduced in the
transmitter. Unlike channel noise, quantization noise
is signal dependent in the sense that it disappears
when the message signal is switched off.
Advantages of PCM
In a generic sense, pulse-code modulation (PCM) has emerged as the most
favored modulation scheme for the transmission of analog information-bearing
signals such as voice and video signals. We may summarize the important
advantages of PCM as follows:
1. Robustness to channel noise and interference.
2. Efficient regeneration of the coded signal along the transmission path.
3. Efficient exchange of increased channel bandwidth for improved signal-to-
noise ratio, obeying an exponential law.
4. A uniform format for the transmission of different kinds of baseband signals,
hence their integration with other forms of digital data in a common network.
5. Comparative ease with which message sources may be dropped or reinserted
in a time-division multiplex system.
6. Secure communication through the use of special modulation schemes or
These advantages, however, are attained at the cost of increased system
complexity and increased channel bandwidth.
T1 Carrier System
The T1 system, which carries 24 voice channels over separate pairs of
wires with regenerative repeaters spaced at approximately 2-km
intervals. each of the 24 voice channels uses a binary code with an 8-bit
word. The first bit indicates whether the input voice sample is positive
(1) or negative (0). The next three bits of the code word identify a
particular segment inside which the amplitude of the input voice sample
lies, and the last four bits identify the actual representation level inside
With a sampling rate of 8 kHz, each frame of the multiplexed signal
occupies a period of 125 μsec. In particular, it consists of twenty-four 8-
bit words, plus a single bit that is added at the end of the frame for the
purpose of synchronization. Hence, each frame consists of a total of
(24 x 8) + 1 = 193 bits. Correspondingly, the duration of each bit equals
0.647 μsec, and the resulting transmission rate is 1.544 megabits per
Time Division Multiplexing (TDM)
The T1 carrier (1.544 Mbps)
Multiplexing T1 streams into higher carriers
T1 (DS1) consists of 24 Telephone calls
T2 (DS2) consists of 24*4= 96 Telephone calls
T3 (DS3) consists of 96*7=672 Telephone calls
T4 (DS4) consists of 672*6= 4032 Telephone calls